Construct the Hamiltonian¶
For representation purposes, we limit ourselves to a rather small lattice size in the 3-dimensional case.
Diagonalization, find eigenvalues, eigenvectors and matrix elements¶
Explanation¶
To extract the array of the matrix elements between eigenstates for the $\hat{n}_i = \hat{c}_i^\dagger\hat{c}_i$ operator at $i$-th site, we first need to extract the $i$-th row of the eigenvector arrays, since the eigenvectors are given as the columns of the said array as per scipy documentation. To obtain the array, we perform a tensor product operation. In our case, we select our site to be located at the center of the array.
Operators to be calculated:¶
Local density¶
$\langle \alpha|\hat{n}_i|\beta\rangle =\langle 0 |\sum_{j,k} \tilde{\alpha}^*_j\tilde{\beta}_k \hat{c}_j\hat{n}_i\hat{c}_k^\dagger|0\rangle = \tilde{\alpha}_i^*\tilde{\beta}_i$
Hopping¶
To obtain the hopping matrix elements, we also extract the $i+m$ row of the eigenvector array. We then take the tensor product of the $i$-th and $(i+m)$-th row and add its conjugate (we also need to ensure all the conjugations in the tensor product are accounted for).
$\langle \alpha|\hat{h}_i^{(m)}|\beta\rangle = \langle \alpha|\hat{c}_i^\dagger\hat{c}_{i+m} + \hat{c}_{i+m}^\dagger\hat{c}_i|\beta\rangle = \tilde{\alpha}^*_{i}\tilde{\beta}_{i+m} + \tilde{\alpha}_{i+m}^*\tilde{\beta}_i$
Densities¶
Hoppings¶
Wavefunctions in the central plane¶
Here, we provide two animations showing the eigenfunctions for the 2D model in the low and high disorder cases. Each frame in the corresponding animations displays one eigenfunction.
